Math 113 (Spring 2009) Yum-Tong Siu 1
Homework Assigned on April 7, 2009
due April 14, 2009
Problem 1 (Solution of the Laplace Equation on the Unit Disk with Neumann
Boundary Condition). For a complex variable ζand another complex variable
zwith |z|<|ζ|, the gradient of log |z−ζ|with respect to zis equal to the
negative of the gradient of log |z−ζ|with respect to ζ. Moreover, by the
Cauchy-Riemann equations for the variable ζ, the outward radial directive of
log |z−ζ|with respect to ζis equal to the tangential derivative of a suitably
defined branch of arg (z−ζ) along the circle of radius |ζ|centered at the
origin in the counterclockwise sense. On the other hand, for |ζ|= 1 the
tangential derivative of arg (z−ζ) with respect to ζalong the unit circle is
related to
lim
β→α
1
β−αarg z−eiβ
z−eiα ,
which, in turn, is related to the Poisson integral kernel constructed by the
method of the argument function and plane Euclidean geometry. This obser-
vation motivates the following way of constructing a solution of the Laplace
equation on the unit disk with Neumann boundary condition.
For zinside the unit disk and for ζon the unit circle, let
Q(z, ζ ) = −2 log |z−ζ|.
Let g(ζ) be a continuous function on the unit circle |ζ|= 1 such that
(∗)Zπ
ϕ=−π
g¡eiϕ¢dϕ = 0.
For z∈Cwith |z|<1, let
u(z) = 1
2πZπ
ϕ=−π
Q¡z, eiϕ ¢g¡eiϕ¢dϕ.
Verify that u(z) is harmonic on the open unit disk {|z|<1}and that for
every fixed ϕ∈Rthe radial derivative
∂
∂r u¡reiϕ ¢
of uapproaches g(eiϕ) as r→1 from r < 1.
Note that the function g(ζ) which defines the Neumann boundary condi-
tion must satisfy (∗) because of the divergence theorem for the gradient of
the harmonic function uon the unit disk.
